the reduction to normal form of a non-normal system of di ...ollivier/production_scient/...reduction...

The reduction to normal form of anon-normal system of differential

equations

De æquationum differentialium systemate nonnormali ad formam normalem revocando

Carl Gustav Jacob Jacobi (1804–1851)Prof. ord. math. Regiom. (1832–1844) & Berol. (1844–1851)

Summarium Abstract

Hanc commentationem in medium pro-tulerunt Sigismundus Cohn, C.W. Bor-chardt et A. Clebsch e manuscriptis po-sthumis C.G.J. Jacobi.

Solutio problematis, datum m2 quan-titatum schema quadraticum per numerosminimos ì singulis horizontalibus adden-dos ita transformandi, ut m maximorumtransversalium systemate præditum eva-dat, determinat systematis m æquatio-num differentialium ui = 0 ordinem etbrevissimam in formam normalem reduc-tionem: æquationes ui = 0 respective ìvicibus differentiandæ sunt.

Etiam indicatur quot vicibus iteratissingulæ æquationes differentiales propo-sitæ differentiandæ sint, ut æquationesdifferentiales nascantur, ad reductionemsystematis propositi ad unicam æquatio-nem necessariæ.

This paper was edited by SigismundCohn, C.W. Borchardt and A. Clebschfrom posthumous manuscripts of C.G.J.Jacobi.

The solution of the following prob-lem: “to transform a square table of m2

numbers by adding minimal numbers ìto each horizontal row, in such a waythat it possess m transversal maxima”,determines the order and the shortestnormal form reduction of the system:the equations ui = 0 must be respec-tively differentiated ì times.

One also determines the number ofdifferentiations of each equation of thegiven system needed to produce the dif-ferential equations necessary to reducethe proposed system to a single equa-tion.

Translated from the Latin byFrançois Ollivier (CNRS)LIX UMR 7161CNRS–École polytechniqueF-91128 Palaiseau CEDEX (France)Email [email protected]

This is the author’s version of the article:Carl Gustav Jacob Jacobi (1804–1851), “The re-duction to normal form of a non-normal systemof differential equations. De æquationum dif-ferentialium systemate non normali ad formamnormalem revocando”, Special issue “Jacobi’sLegacy” of AAECC, J. Calmet and F. Olliviereds, 20, (1), , 33–64, 2009.DOI 10.1007/s00200-009-0088-2

1

2 C.G.J. Jacobi

Translator’s comments

T his text [Jacobi 2] is based on a single manuscript [II/23 b)], thetranscription of which was considered and started by Cohn in 1859(see his letter to Borchardt [II/13 a)] and Borchardt’s abstract [II/25]that refers to his and Cohn’s transcription). The first published version ap-peared a volume edited by Clebsch [VD]. Nothing indicates his participationin the preparation of the text. One may guess that the main part of the work,if not all, was achieved by Borchardt after Cohn’s death, but the transcrip-tions are lost. A strange manuscript [II/24], from some unknown hand, mixesthe texts of the two articles related to Jacobi’s bound [Jacobi 1 and Jacobi2], showing that their publication in a single paper has been considered.

This published version follows closely the original texts, and most changesonly concern questions of style that are not perceptible in the translation,e.g. permuation of words, the order in Latin sentences being very flexible.The original text is in general easier to translate. Having translated thepublished paper before I have found the original manuscript, I often had tosimplify it to make the translation readable: many times these changes justrestored Jacobi’s version. Additions made by Borchardt were kept when theycan help the reader. [They are indicated by sans sherif letters enclosed betweenbrackets].

Before the translation of the published article [Jacobi 2], we provide thesecond part of an unpublished manuscript [II/23 a), fos 2217–2220], relatedto the different normal forms that a given system may admit, and the wayto compute them, that may help to understand the mathematical context.This text was possibly intended as a part of an article [Jacobi 3], but the§ 17 of the published version only considers changes of normal forms for linearsystems.

Thanks are due to Alexandre Sedoglavic for his kind help and encourage-ments to achieve a preliminary French translation. I must express my mostheartfelt thanks to Daniel J. Katz, who also corrected the translation of acompanion paper [Jacobi 1]. He provided again a very competent rereadingof the latin transcription and of this translation, suggesting many importantimprovements and correcting numerous mistakes and inaccuracies.

References

Primary material, manuscripts

The following manuscripts from Jacobis Nachlaß, Archiv der Berlin-Branden-burgische Akademie der Wissenschaft, were used to establish this translation.

Reduction to normal form of a non-normal system of differential equations 3

We thank the BBAW for permission to use this material and its staff for theirefficiency and dedication.

[II/13 a)] Letter from Sigismund Cohn to C.W. Borchardt. Hirschberg,August, 25th 1859, 3 p.

[II/23 a)] Reduction simultaner Differentialgleichungen in ihre canonischeForm und Multiplicator derselben, manuscript by Jacobi. Five differentfragments: 2214–2216; 2217–2220 (§ 17-18); 2221–2225 (§ 17); 2226–2229;2230–2232, 2235, 2237, 2236, 2238.

[II/23 b)] De aequationum differentialium systemate non normali ad formamnormalem revocando, manuscript by Jacobi 2238–2251. The basis of [Ja-cobi 2].

[II/24] De aequationum differentialium systemate non normali ad formamnormalem revocando, manuscript from unknown hand, mixing [Jacobi 1]and [Jacobi 2].

[II/25] De aequationum differentialium systemate non normali ad formamnormalem revocando. Abstract and notes by Borchardt. 8 p.

Publications

[Crelle 27] Journal für die reine und angewandte Mathematik, 27, Berlin,Georg Reimer, 1844.

[Crelle 29] Journal für die reine und angewandte Mathematik, 29, Berlin,Georg Reimer, 1845.

[GW IV] C.G.J. Jacobi’s gesammelte Werke IV, K. Weierstrass ed., Berlin,Georg Reimer, 1886.

[GW V] C.G.J. Jacobi’s gesammelte Werke V, K. Weierstrass ed., Berlin,Georg Reimer, 1890.

[VD] Vorlesungen über Dynamik von C. G. J. Jacobi nebstes fünf hinter-lassenen Abhandlungen desselben, A. Clesch ed., Berlin, Georg Reimer,1866.

[Jacobi 1] Jacobi (Carl Gustav Jacob), “De investigando ordine systematisaequationum differentialum vulgarium cujuscunque”, [GW V] p. 193-216,and this special issue of AAECC.

[Jacobi 2] Jacobi (Carl Gustav Jacob), “De aequationum differentialum sys-temate non normali ad formam normalem revocando”, [VD] 550–578 and[GW V] 485-513.

[Jacobi 3] Jacobi (Carl Gustav Jacob), “Theoria novi multiplicatoris sys-temati aequationum differentialium vulgarium applicandi”, [Crelle 27] 199–268, [Crelle 29] 213–279 and 333–376; [GW IV], 317–509.

Translations

[II/23 a) fos 2217–2220]

The transformations of a system ofdifferential equations of an arbitrary order;

how the multiplier of the transformedsystem is obtained from the multiplier of the

proposed system.17.

The system of differential equations1)

dn1x1dtn1

= A1,dn2x2dtn2

= A2, . . . ,dnmxmdtnm

= Am

is proposed, in which only derivatives lower than dn1x1dtn1

, dn2x2dtn2

etc. are to befound on the right side. The order of this system is the number

n1 + n2 + · · ·+ nmthat is the sum of the orders n1, n2, . . . , nm, up to which go the derivativesof each variables in the proposed equations. One agrees that the order ofan arbitrary system of differential equations is defined as being equal to thenumber of first order differential equations to which it may be reduced, oralso to the number of arbitrary constants that its complete integration admitsand requires1. The proposed system of differential equations can always betransformed into some others of the same order, and the order up to whichthe derivatives of each variable in the transformed equations must go willbe, in general, left to our discretion, provided that their sum be equal to thesystem order. Such a transformation is done in the following way.

It is proposed to transform the system of equations 1) into some other,in which the derivatives of variables x1, x2, . . . , xm respectively reachthe orders i1, i2, . . . , im, where

i1 + i2 + · · ·+ im = n1 + n2 + · · ·+ nm.Always denoting the derivatives by upper indices in Lagrange’s way, the givensystem may be reduced to differential equations of the first order betweenthe variables

1I do not say to the order of a differential equation between two of the proposed vari-ables, to which the proposed system may be reduced by appropriate differentiations andeliminations; indeed, in some particular cases this reduction does not succeed, for exampleif A1 is a function of x1 and t alone, A2 of x2 and t alone, . . . , Am of xm and t alone. J.

5

6 C.G.J. Jacobi

A)

t, x1, x

′1, x′′1, . . . , x

(n1−1)1 ,

x2, x′2, x′′2, . . . , x

(n2−1)2 ,

. . . . . . . . . . . . . . .xm, x

′m, x

′′m, . . . , x

(nm−1)m .

In the same way, the system that it is proposed to obtain by this transforma-tion is equivalent to a system of first order equations between the variables

B)

t, x1, x

′1, x′′1, . . . , x

(i1−1)1 ,

x2, x′2, x′′2, . . . , x

(i2−1)2 ,

. . . . . . . . . . . . . . . . .xm, x

′m, x

′′m, . . . , x

(im−1)m .

So, the problem is reduced to transforming a system of first order differentialequations between the variables A) into some other one between the variablesB).

Let us assume that

the orders i1, i2, . . . , ik are resp. greater than n1, n2, . . . nk

˝ ik+1, ik+2, . . . , ih−1 ˝ equal to nk+1, nk+2, . . . nh−1

˝ ih, ih+1, . . . , im ˝ smaller than nh, nh+1, . . . nm;

We shall need to replace the variables

C)

x

(ih)h , x

(ih+1)h , . . . , x

(nh−1)h ,

x(ih+1)h+1 , x

(ih+1+1)h+1 , . . . , x

(nh+1−1)h+1 ,

. . . . . . . . . . . . . . . . . . . . . . .

x(im)m , x(im+1)m , . . . , x

(nm−1)m

by these new ones,

D)

x

(n1)1 , x

(n1+1)1 , . . . , x

(i1−1)1 ,

x(n2)2 , x

(n2+1)2 , . . . , x

(i2−1)2 ,

. . . . . . . . . . . . . . . . . .

x(nk)k , x

(nk+1)k , . . . , x

(ik−1)k .

In order to obtain the substitutions requested for that goal, I differentiaterepeated times the equations

x(n1)1 = A1, x

(n2)2 = A2, . . . , x

(nk)k = Ak

and as soon as the nthµ derivative of the variable xµ appears in the rightpart, I substitute for it its value Aµ from 1). As the given functions A1,A2, . . . , An only involve the variables A), the quantities D) are in this wayalso all equal to functions of A) only. I will call these equations, providing


the necessary substitutions, auxiliary equations. In order that the proposedtransformation be possible, it must be that the values of the variables C)to eliminate could be reciprocally obtained from the auxiliary equations bywhich the values of the new variables D) are determined2, so that, usingthe auxiliary equations, the quantities D) are expressed by A) or also thequantities C) by B). Differentiating again the values of quantities

x(i1−1)1 , x

(i2−1)2 , . . . , x

(ik−1)k ,

expressed by variables A), and substituting the proposed differential equa-tions 1), the values of the quantities

x(i1)1 , x

(i2)2 , . . . , x

(ik)k

appear, expressed by variables A). These, expressed by variables B) with thehelp of the auxiliary equations, become

B1, B2, . . . , Bk.

Then, let the quantities

Ak+1, Ak+2, . . . , Ah−1x

(ih)h , x

(ih+1)h+1 , . . . , x

(im)m ,

be respectively changed, with the help of the auxiliary equations, in functionsof the variables B):

Bk+1, Bk+2, . . . , Bh−1Bh, Bh+1, . . . , Bm;

one will have the system of transformed differential equations,

2)di1x1dti1

= B1,di2x2dti2

= B2, . . . ,dimxmdtim

= Bm,

where the functions B1, B2, . . . , Bm placed on the right contain only deriva-tives smaller than those on the left.

The fact that one can get back to the proposed equations from the trans-formed differential equations by differentiations alone, appears by represent-ing the systems 1) and 2) as systems of differential equations of the first orderrespectively between the variables A) and bewteen the variables B). In thecase of first order equations, the way for going back from the transformeddifferential equations to the proposed ones is in fact straightforward. Indeed,n differential equations between t, x1, x2, . . . , xn, of the following form

3) ui =dxidt−Xi = 0,

2Hence, e. g., the values of quantities D) must involve all the variables C). J.

8 C.G.J. Jacobi

are transformed into some others, between the variables t, y1, y2, . . . , yn,denoting by y1, y2, . . . , yn given functions of the variables t, x1, x2, . . . , xn.Taking

4) Yi =∂yi∂t

+∂yi∂x1

X1 +∂yi∂x2

X2 + · · ·+∂yi∂xn

Xn,

gives the transformed differential equations

5) 5) vi =dyidt− Yi = 0,

where quantities Y1, Y2, . . . , Yn are expressed by the variables t, y1, y2, . . . ,yn. To go back from these equations to the proposed ones, one needs torestore in the place of each yi, the functions to which they are equal, fromwhich the quantities Yi are changed to the values 4); this being done, as

dyidt

=∂yi∂t

+∂yi∂x1

dx1dt

+∂yi∂x2

dx2dt

+ · · ·+ ∂yi∂xn

dxndt

,

the equation 5), if we again set ui =dxidt−Xi, is changed in the following,

6) 6) 0 =∂yi∂x1

u1 +∂yi∂x2

u2 + · · ·+∂yi∂xn

un.

Assigning to the index i the values 1, 2, . . . , n one obtains from the precedingequation n equations, linear in each of the u1, u2, . . . , un and deprived ofconstant terms; whenever the determinant of these equations∑

±∂y1∂x1

∂y2∂x2· · · ∂yn

∂xn

does not vanish, one necessarily has

u1 = 0, u2 = 0, . . . , un = 0,

which is the proposed system of differential equations. And it is known thatthis determinant does not vanish whenever y1, y2, . . . , yn be independentfunctions of x1, x2, . . . , xn, that is whenever that one can, from the values ofthe quantities y1, y2, . . . , yn, expressed by t, x1, x2, . . . , xn get reciprocallythe values of the quantities x1, x2, . . . , xn, expressed by t, y1, y2, . . . , yn, oralso whenever that no equation can appear between t, y1, y2, . . . , yn alone.Such a supposition is implicitely understood, when considering to transformn differential equations between t, x1, x2, . . . , yx into some others betweent, y1, y2, . . . , yn.

In the same way, one may go back to the proposed ones from the dif-ferential equations 2) every time that the auxiliary equations, by which thevariables D) are expressed by A), are such that one can obtain from themthe values of the variables C), which was requested above, or equivalently


such that one cannot get from them an equation between the quantities B)alone. This is a necessary and sufficient condition for having the proposedtransformation. And, this transformation being done, going back from thetransformed equations 2) to the proposed ones, may be conceived in this way:the equations 2) are first reduced to a system of differential equations of thefirst order, that will be between the variables B), and then, without consid-ering their signification as derivatives, the new quantities C) are introducedin it, at the place of the quantities D), using the same auxiliary equations bywhich the transformation was done.

Hence, one will get a system of first order equations between the quantitiesA), which shows that these quantities are equal to the derivatives expressedby their upper indices and that substituting for the quantities A) the deriva-tives to which they are equal provides the proposed differential equations 1)themselves.

Let ∆ be the determinant of the quantities D) formed with respect to thequantities C), which according to what precedes cannot vanish, so that theproposed tranformation may be achieved, let then M and N be the multi-pliers of the proposed 1) and transformed 2) differential equations. We haveaccording to § 93

M = ∆.N ;

then according to formula 5)4 § 143 dLgMdt

and dLgNdt

are defined by the formulæ,

dLgM

dt= −

{∂A1

∂x(n1−1)1

+∂A2

∂x(n2−1)2

+ · · ·+ ∂Am∂x

(nm−1)m

}dLgN

dt= −

{∂B1

∂x(i1−1)1

+∂B2

∂x(i2−1)2

+ · · ·+ ∂Bm∂x

(im−1)m

}.

This provides the memorable formula,

7)

∫ { ∂A1∂x

(n1−1)1

+∂A2

∂x(n2−1)2

+ · · ·+ ∂Am∂x

(nm−1)m

}dt

=∫ { ∂A1

∂x(i1−1)1

+∂A2

∂x(i2−1)2

+ · · ·+ ∂Am∂x

(im−1)m

}dt+ Lg∆,

by which one integral is reduced to the other.

3This numeration is compatible with that of [Jacobi 3]. T.N.4The formula number is missing in [GW IV] but appears in [Crelle 29]. T.N.

10 C.G.J. Jacobi

18.

The equations, from which by suitable eliminations the proposed transfor-mation was obtained, were formed by differentiiating repeatedly some par-ticular differential equations in 1) and when, just after each differentiation,some derivative xnµµ appeared in their right side, substituting its value Aµ.But it is sometimes interesting to separate the tasks of differentiations andsubstitutions, so that only by differentiations of the proposed differentialequations, without doing any substitutions, one forms a system of equations,from which the transformed differential equations may then be obtained bysubstitutions or eliminations alone, without doing any more differentiations.If one wishes to perform the transformation in that way, this problem is theforemost to be solved

“if one differentiates one of the equations 1), e.g. x(n1)1 = A1, f1 times,

to differentiate all the others in order to obtain all the equations, inwhich appears no derivative of the variables x2, x3, . . . , xm greaterthat those that appear in the equation

df1x(n1)1

dtf1=df1A1dtf1

.”

One may conceive the solution of this problem in the following way.

In the proposed differential equations 1) and in the others that come fromtheir differentiation, we decide to denote the derivatives of x1 up to the n1

th,of x2 up to the n2

th, . . . , of xm up to the nmth in Lagrange’s way using

indices; but to denote the greater derivatives that appear differentiating theequations 1) by the usual symbol d. Using this notation in forming the

expression df1A1dtf1

, I look for the highest derivative of the quantities

x(n2)2 , x

(n3)3 , . . . , x

(nm)m .

that appears in it. When it is the f th2 of x(n2)2 , I differentiate

x(n2)2 = A2,

f2 times and look for the highest derivative of

x(n3)3 , x

(n4)4 , . . . , x

(nm)m .

that appears in the expressions

df1A1dtf1

,df2A2dtf2

.


When it is the f3th of x

(n3)3 , I look again for the highest derivative of x

(n4)4 ,

x(n5)5 etc. appearing in the expressions

df1An11dtf1

,df2A2dtf2

,df3A3dtf3

.

This being posed, defining new differentiations, we achieve in this way thewhole system of differential equations that was requested in the proposedproblem

8)

x(n1)1 = A1,

dx(n1)1

dt=dA1dt

,d2x

(n1)1

dt2=d2A1dt2

, . . . ,df1x

(n1)1

dtf1=df1A1dtf1

,

x(n2)2 = A2,

dx(n2)2

dt=dA2dt

,d2x

(n2)2

dt2=d2A2dt2

, . . . ,df2x

(n2)2

dtf2=df2A2dtf2

,

etc. etc.

x(nm)m = Am,dx(nm)mdt

=dAmdt

,d2x(nm)mdt2

=d2Amdt2

, . . . ,dfmx(nm)mdtfm

=dfmAmdtfm

.

I observe that, if, when searching for the highest derivative, many highestderivatives of the same order are to be found, the way of proceeding does notchange. Let, e.g., the highest derivatives of the two quantities x

(n2)2 and x

(n3)3

in the expression df1A1dtf1

be of the same order f2, the highest derivatives ofthe remaining variables x4, x5 etc. being of a smaller order; in the expressiondf2A2dtf2

, the hightest derivatives of the quantities xn33 , xn44 , . . . , x

(nm)m will be

of a smaller order than the f th2 , so that the fth2 derivative of the quantity

x(n3)3 will the highest derivative among those of the quantities x

(n3)3 , x

(n4)4 etc.

appearing in the two expressions df1A1dtf1

and df2A2dtf2

. For that reason, if among

all the derivatives of the quantities x(n2)2 , x

(n3)3 , . . . , x

(nm)m appearing in the

expression df1A1dtf1

, two highest f th2 derivatives of the quantities x(n2)2 and x

(n3)3

are found, we may continue like this, looking in the expressions

df1A1dtf1

,df2A2dtf2

,df2A3dtf2

for all the derivatives of the quantities x(n4)4 , x

(n5)5 etc. that are of the highest

order. One proceeds in the same way if in observing the highest order ofderivatives, many are found to be of the same highest order. The numbersf1, f2, . . . , fm decrease continuously, but so that exceptionally some may bemutually equal.

These preliminaries done, let us now investigate how many times each of theproposed equations must be differentiated to obtain a system of differentialequations from which the transformed differential equations 2) may appear.Let us assume again that the orders i1, i2, . . . , ik are respectively greater than

12 C.G.J. Jacobi

n1, n2, . . . , nk, the remaining ik+1, ik+2 etc. respectively equal or smaller thannk+1, nk+2 etc. I take the maximum of the numbers

i1 − n1, i2 − n2, . . . , ik − nk;

when it is

i1 − n1 = f1I form the system of equations 8), according to the given method. When inthem

f2 ≥ i2 − n2, f3 ≥ i3 − n3, . . . , fk ≥ ik − nk,

what was proposed will be satisfied. In the contrary case, among the numbersi2 − n2, i3 − n3, . . . , ik − nk that are respectively greater than the numbersf2, f3, . . . , fk, I look for the maximum. Let it be e.g. i4 − n4, differentiatingagain the proposed differential equations 1), I form by the exposed methodthe system of all the equations in which one finds no derivative of the variablesx2, x3, x5, . . . , xm, greater than those that appear in the equation

di4x4dti4

=di4−n4A4dti4−n4

.

I repeat the same operation until in the formed systems of equations, theorders up to which the derivatives of x1, x2, . . . , xk go are respectively equalor greater than i1, i2, . . . , ik. By this method, as I produce the appropriatenumber of equations from which the transformed differential equations 2)may arise by simple eliminations, then all the equations that are formed bythe different differentiations will be necessary to perform these eliminations.We may at the same time dispose the equations in such a order that thequantities appearing on the left of each equation do not appear in all thepreceding equations. These quantities are the derivatives of the variables x1,x2, . . . , xm respectively greater than the (n1−1)th, (n2−1)th, . . . , (nm−1)th.When the equations are disposed in this order, they are easilly determined bylower derivatives, that is by quantities A), as each equation, by substitutionsof the equations preceding it alone, provides at once the determination of thenew quantities. In this way, when the quantities D) by [. . . ]5

5The sentence is interrupted at the bottom of the page. T.N.

[Jacobi 2]

[The reduction in normal form of a non normalsystem of differential equations]

The multiplier of a system of differential

equations not being in normal form

L ooking for the multiplier of isoperimetrical differential equa-tions creates much greater difficulties if the highest derivativesof the functions x, y, z, etc. appearing in the expression U are not ofthe same order.6 In this case, the isoperimetrical system of differentialequations will not be, as I have always assumed up to now, in sucha form that the highest derivative of each dependent variables couldbe taken as unkowns, the values of which will be determined by theequations themselves.

We reduce then the isoperimetrical differential equations to theform that I have indicated only after some differentiations and elimi-nations; this makes complicated the search for the value of the mul-tiplier. As a reward for this work, I have obtained all the necessarymaterial to expose with care the reduction in normal form of a nonnormal system of differential equations. In this search, I came to gen-eral propositions that one will see to fill some gap in the theory ofordinary differential equations, a summary of which I will indicatehere briefly.

6See [GW IV, p. 395]. T.N.

13

14 C.G.J. Jacobi

[§. 1.The order of a system of m differential equations andits fastest normal form reduction are determined by theresolution of the following problem: to transform a givensquare table of m2 quantities by adding to each line min-imal numbers `1, `2, . . . , `m in order to equip it with asystem of transversal maxima.

The resolution is illustrated with an example.]

W e call again the independent variable t, its functions, or variablesconsidered as dependent, x1, x2, . . . , xm. Let there be m differentialequations between these variables:

u1 = 0, . . . , um = 0.

Let ai,κ be the highest order of a derivative of variable xκ in equation ui = 0.I say that:

1) the order of the proposed system of differential equations, or equiva-lently the number of arbitrary constants that their complete integrationrequires, is equal to the maximum of all the sums:

ai1,1 + ai2,2 + · · ·+ aim,m,

if we choose the indices i1, i2, . . . , im all different the one from the other,in all possible ways, among the indices 1, 2, . . . ,m.

I will denote by O this maximum, that is the system order; O will be equalto the sum of orders of the highest derivatives of each variable appearing ina normal system to which the proposed system may be reduced, a sum thatin the proposed system of differential equations is greater than O.

There exist different normal forms, and always at least two, to which thesystem may be reduced, reductions that require the help of various differ-entiations and eliminations. In this field, this proposition is fundamental,

2) among the various ways to differentiate the proposed differential equa-tions to obtain the auxiliary equations with the help of which the pro-posed system may be reduced to normal form by eliminations alone,there exists a unique way that requires the least number of differentia-tions, meaning that, by any other way, some of the proposed differentialequations, or all of them, must be differentiated a greater number oftimes, and none can be differentiated a smaller number of times.


We denote this fastest way under the name of shortest reduction. Inthis reduction, there will always be one or more of the proposed differentialequations that will not be differentiated at all, meaning that none of itsderivatives contributes to the system of auxiliary equations. So, if we assumethat in order to form the auxiliary equations, the equation ui = 0 must bedifferentiated `i successive times, among the positive integers :

`1, `2, . . . , `m

one or more must be equal to zero. In order to find these numbers, on whichfully depends the shortest reduction, one must solve this problem.

Problem.

“Given m2 arbitrary quantities ai,κ, where i and κ must take the val-ues 1, 2, . . . ,m, let ai,κ + `i = pi,κ; to search for m minimal positive orzero quantities `1, `2, . . . , `m such that we may choose, among the m

2

quantities pi,κ, m quantities :

pi1,1, pi2,2, . . . , pim,m

placed in different horizontal and vertical series, each of which takingthe maximal value among the quantities of the same vertical or beingless than no other quantity of the same vertical.”

Solution.

I will briefly indicate the main steps of the solution of the given problem.Let us arrange the quantities ai,κ in a square table:

A. a1,1, a1,2, . . . , a1,m,a2,1, a2,2, . . . , a2,m,

. . . . . . . . . . . . . . . . . . . . .am,1, am,2, . . . , am,m.

If we find a horizontal series of which no term is maximal (by this word,I will always understand that it is not less than any of the others terms)among all those of the same vertical, I add to all the terms of this horizontalseries the same positive quantity of minimal value such that one of its termsbecome equal to the maximum of its vertical.

After the indicated preparation, if we have changed the given square tothe following,

B. b1,1, b1,2, . . . , b1,m,b2,1, b2,2, . . . , b2,m,. . . . . . . . . . . . . . . . . . . . .bm,1, bm,2, . . . , bm,m,

16 C.G.J. Jacobi

there will be no horizontal series of the square B, in which there is no maximalterm among all these of its vertical. I associate to such a square the followingdefinitions that are to be well remembered.

I call a system of transversal maxima a system of quantities bi,κ that areall placed in different horizontal and vertical series and maximal among allthe terms placed in the same vertical.

I take in the square B the maximal number of transversal maxima and,when the same number of transversal maxima may be produced in manyways, I chose arbitrarily one of these systems, the terms of which I markwith an asterisk. The maximal number of these transversal maxima may beor 1, or 2 7 etc. or m ; if their number is m, the given problem is solved.If this number is smaller than m, I make it so that some horizontal seriesare increased by minimal numbers, such that a greater number of transversalmaxima is found in the obtained new square. Repeating this process, it isnecessary that one gets a square in which the number of transversal maximais m, and at this stage the solution of the problem is found. I say here thata horizontal series is increased, if the same positive quantity is added to allits terms.

I call respectively H and V the horizontal and vertical series to which thechosen system of transversal maxima belongs and the other horizontal andvertical series H ′ and V ′. I also mark with an asterisk the maximal terms ineach vertical of V ′8. I call starred maxima the terms marked with an asterisk.

Let us assume that in the horizontal series h1 there is a starred maximumto which is equal a term of some horizontal series h2 placed in the samevertical; that in the series h2 there is a starred maximum to which is equala term of the same vertical placed in some horizontal series h3, [. . . ] etc. Ifin this way we reach the horizontal series hα, where hα denotes one of theseries h2, h3, . . . , hm, I will say that there is a path from h1 to hα. If one saysthat there is a path from h1 to hα, the series h2, h3, . . . , hα−1 belong to theseries of H; the series hα may belong to the series of H or to those of H

′.If there is no path to a series of H ′ from a horizontal series in which thereare two or more starred maxima and if there is no maximal term of some

7[The applied preparation, by which the square table A has been changed in the table B,makes 2 the minimal value of this number, this value appearing if all the maxima are placedin the same horizontal series and if, moreover, in some vertical all the terms are equal the onewith the other. See [Jacobi 1].]

8A litteral reading of the original text terminos in una verticalium V ′ is maximal termsin one of the verticals of V ′. But a correct description of the algorithm requires to markall such terms. T.N.


series of H ′ in some series of V ′, this is a criterion certifying that a maximalnumber of transversal maxima has been chosen.

This being posed, I distribute all the horizontal series in three classes.I associate to the first class of horizontal series the series in which

one finds two or more starred maxima, and also all the horizontal seriesto which there is a path from these series; no series of the first classbelong to H ′.

I associate to the second class of horizontal series the series of Hnot belonging to the first class, from which there is no path to a seriesof H ′.

I associate to the third class of horizontal series all the series of H ′

and the series of H from which there is a path to some series of H ′.

The partition of horizontal series being made, I increase all the series belong-ing to the third class of a same quantity, the smallest such that one term ofthese series become equal to some starred maxima of the first or second classplaced in the same vertical. If this starred maximum belongs to some hori-zontal series of the second class, this one, in the new square obtained, goesto the third class and nothing else changes in the series repartition9. In thiscase, the operation must be iterated, the new series being transfered fromthe second to the third class, until one term of the series of the third classbecome equal to some starred maxima of a series in the first class. If it doesnot happen sooner, this will necessarily happen when all the second class se-ries are gone to the third. And we obtain at the same time a square in whichthere is a greater number of transversal maxima than in the square B. Then,with a new disposition of the starred maxima and a new distribution of thehorizontal series in three classes, a new square must be formed by the samemethod, in which the number of transversal maxima will be increased again;one must go on until one reaches a square in which there are m transversalmaxima. The square found in this way will be obtained from the proposedsquare A by adding to the horizontal series minimal positive quantities thatwill be the requested `1, `2, . . . , `m.

Due to the complexity of the rule, it is convenient to present a single example,contained in the following figure. The proposed square itself is A. I haveunderlined the terms in it which are maximal in their vertical, and havedone likewise for the square derived from it. I have denoted the horizontalseries by the letters a, b, . . . , k. Among them, b, c, e, f, i, k do not contain any

9Up to the fact that all the series from which there is a path to this second class serieswill go to the third class with it. T.N.

18 C.G.J. Jacobi

The eight tables of the example appear on fo 2250v; the recto contains the list of the doubles of prime numbers congruent to1 modulo 8, from 2018 to 20018. A marginal note of Jacobi in German in the margin of fo 2239v indicates that they mustappear in this part of the text. T.N.

A. E.

α β γ δ ε ζ η ϑ ι κ

a 14 23 1 5 73 91 10 34 5 99

b 25 32 2 4 62 81 9 23 4 88

c 14 1 7 16 21 7 13 12 3 77

d 11 53 61 4 3 1 12 1 4 91

e 9 21 23 18 27 3 6 9 12 15

f 4 16 18 13 5 12 23 21 14 81

g 25 43 13 16 83 10 91 3 7 13

h 27 7 17 37 73 8 11 24 23 22

i 25 12 18 27 32 18 24 23 14 88

k 16 28 30 25 34 10 13 16 19 42

V V V ′ V V ′ V V V V V

III a 14 23 1 5 73 91 10 34 5 99∗

III b 35 42 12 14 72 91∗ 19 33 14 98

III c 36∗ 23 29 38 43 29 35 34 25 99

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 29 41 43 38 47 23 26 29 32∗ 35

III f 17 29 31 26 18 25 36 34∗ 27 94

I g 25 43 13 16 83∗ 10 91∗ 3 7 13

II h 28 8 18 38∗ 74 9 12 25 24 23

III i 36 23 29 38 43 29 35 34 25 99

III k 29 41 43 38 47 23 26 29 32 55

11 18 9 55

B. F.V V V ′ V V ′ V ′ V V ′ V V

I a 14 23 1 5 73 91∗ 10 34∗ 5 99∗

III b 27∗ 34 4 6 64 83 11 25 6 90

III c 27 14 20 29 34 20 26 25 16 90

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 20 32 34 29 38 14 17 20 23∗ 26

III f 13 25 27 22 14 21 32 30 23 90

I g 25 43 13 16 83∗ 10 91∗ 3 7 13

II h 27 7 17 37∗ 73 8 11 24 23 22

III i 27 14 20 29 34 20 26 25 16 90

III k 20 32 34 29 38 14 17 20 23 46

19 27 8 19 8 59 4 9

V V V ′ V V V V V V V

III a 23 32 10 14 82 100 19 43 14 108∗

III b 44 51 21 23 81 100∗ 28 42 23 107

III c 45∗ 32 38 47 52 38 44 43 34 108

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 38 50 52 47 56 32 35 38 41∗ 44

III f 26 38 40 35 27 34 45 43∗ 36 103

II g 25 43 13 16 83 10 91∗ 3 7 13

II h 37 17 27 47 83∗ 18 21 34 33 32

III i 45 32 38 47∗ 52 38 44 43 34 108

III k 38 50 34 47 56 32 35 38 41 64

2 9 1 46

C. G.V V V ′ V V ′ V ′ V V V V

I a 14 23 1 5 73 91∗ 10 34 5 99∗

III b 31∗ 38 8 10 68 87 15 29 10 94

III c 31 18 24 33 38 24 30 29 20 94

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 24 36 38 33 42 18 21 24 27∗ 30

III f 17 29 31 26 18 25 36 34∗ 27 94

I g 25 43 13 16 83∗ 10 91∗ 3 7 13

II h 27 7 17 37∗ 73 8 11 24 23 22

III i 31 18 24 33 38 24 30 29 20 94

III k 24 36 38 33 42 18 21 24 27 50

15 23 4 15 4 61 5 5

V V V ′ V V V V V V V

III a 24 33 11 15 83 101 20 44 15 109∗

III b 45 52 22 24 82 101∗ 29 43 24 108

III c 46∗ 33 39 48 53 39 45 44 35 109

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 39 51 53 48 57 33 36 39 42∗ 45

III f 27 39 41 36 28 35 46 44∗ 37 104

II g 25 43 13 16 83 10 91∗ 3 7 13

II h 37 17 27 47 83∗ 18 21 34 33 32

III i 46 33 39 48∗ 53 39 45 44 35 109

III k 39 51 35 48 57 33 36 39 42 65

1 8 45

D. H.V V V ′ V V ′ V V V V V

II a 14 23 1 5 73 91 10 34 5 99∗

II b 35 42 12 14 72 91∗ 19 33 14 98

III c 35∗ 22 28 37 42 28 34 33 24 98

I d 11 53∗ 61∗ 4 3 1 12 1 4 91

III e 28 40 42 37 46 22 25 28 31∗ 34

II f 17 29 31 26 18 25 36 34∗ 27 94

I g 25 43 13 16 83∗ 10 91∗ 3 7 13

III h 27 7 17 37∗ 73 8 11 24 23 22

III i 35 22 28 37 42 28 34 33 24 98

III k 28 40 42 37 46 22 25 28 31 54

13 19 10 63 57 1 1


S3 a 25 34 12 16 84 102∗ 21 45 16 110

S2 b 46 53∗ 23 25 83 102 30 44 25 109

S5 c 47∗ 34 40 49 54 40 46 45 36 110

S1 d 11 53 61∗ 4 3 1 12 1 4 91

S6 e 40 52 54 49 58 34 37 40 43∗ 46

S4 f 28 40 42 37 29 36 47 45∗ 38 105

S1 g 25 43 13 16 83 10 91∗ 3 7 13

S4 h 38 18 28 48 84∗ 19 22 35 34 33

S4 i 47 37 40 49 54 40 46 45 36 110

S5 k 40 52 36 49∗ 58 34 37 40 43 66


underlined term. Subtracting the terms of b of the underlined terms of theirverticals, one gets the differences

2, 21, 59, 33, 21, 10, 82, 11, 19, 11,

of which 2 is the smallest, so I increase by 2 the series b. The terms of theseries c respectively differ from the underlined terms of their verticals by thequantities

13, 52, 54, 21, 62, 84, 78, 22, 20, 22;

as 13 is the smallest of them, I increase the series c of the quantity 13. Inthe same way, I deduce the square B by increasing e, f, i, k of the respectivequantities 11, 9, 2, 4 and I denote its construction by the symbol:

B. (a, b+ 2, c+ 13, d, e+ 11, f + 9, g, h, i+ 2, k + 4).

In the square B, one can assign 6 transversal maxima and not more; the ver-tical series in which they are placed are surmounted with a V , the remainingwith a V ′. I denote these maxima with an asterisk. If one finds an under-lined term in some series of V ′, I denote it also with an asterisk. I attachthe horizontal series a, d, g, in which two or more starred maxima are found,to the class I. In the 7 verticals to which belong these maxima, on finds noother underlined term, so there is no path from one of these series to someother and a, d and g constitute alone the first class. The series c, f, i, k, asone finds in them no starred term, belong to class III. Then, there is a pathfrom e to the series f and k and from b to c and i; so the series b and e alsobelong to the third class. For, according to the definition above, there is apath to a horizontal series s from some other s1, if there is in s an underlinednon-starred term and in the same veritical a starred term belonging to theseries s1. As the series a, d, g belong to the first and the series b, c, e, f, i, k tothe third class, the series h remains that constitutes the second class. Now,in each vertical series in which there is a starred maximum of a first or sec-ond class series, one takes a nearest lower term of a series of the third classand one notes the difference of the two terms under the vertical series. Fromthese differences:

53− 34 = 19, 61− 34 = 27, 37− 29 = 8, 83− 64 = 19,91− 83 = 8, 91− 32 = 59, 34− 30 = 4, 99− 90 = 9,

one takes the smallest 4 ; one obtains the next square by increasing all theseries of the third class of the same quantity 4. This square may be denotedby the symbol

C. (a, b+ 6, c+ 17, d, e+ 15, f + 13, g, h, i+ 6, k + 8).

We see that in the square C, one finds 7 transversal maxima and that a newstarred term appeared in series f ; this series goes from the third to the second

20 C.G.J. Jacobi

class. I write below the quantities by which the starred terms belonging toseries of the first and second class dominate in the square C the nearest lower[third class] terms of the same vertical. As the smallest of these quantitiesis 4, augmenting by 4 all the series of class III, I form the square

D. (a, b+ 10, c+ 21, d, e+ 19, f + 13, g, h, i+ 10, k + 12),

in which there are already 8 transversal maxima. According to the givenrules, the disposition of asterisks must be modified a little in the square D;this being done, the series a, b and h are found to move from classes I, IIIand II to classes II, II and III. The starred terms of classes I and II exceedthe immediately lower terms [of class III and] of the same verticals by thenumbers 13, 19, 10, 63, 57, 1, 1; augmenting by their minimum 1 all theseries of class III, I deduce the square

E. (a, b+ 10, c+ 22, d, e+ 20, f + 13, g, h+ 1, i+ 11, k + 13),

in which the number of transversal maxima is the same. The structure ofsquare E does not differ from that of the square D except in the fact thatthree class II series a, b and f moved to class III. Viz., f and a moved toclass III because their starred terms 34 and 99 are equal to the terms of theseries i and c placed in the same verticals; regarding b, it moved to class IIIbecause its starred term 91 is equal to the term of the series a in the samevertical, that already went to class III. From square E, one deduces by theenunciated rules the square

F. (a+ 9, b+ 19, c+ 31, d, e+ 29, f + 22, g, h+ 10, i+ 20, k + 22),

in which there are 9 transversal maxima; from the square F , one deduces thesquare

G. (a+ 10, b+ 20, c+ 32, d, e+ 30, f + 23, g, h+ 10, i+ 21, k + 23),

in which there are also 9 transversal maxima; at last, from G comes therequested square

H. (a+ 11, b+ 21, c+ 33, d, e+ 31, f + 24, g, h+ 11, i+ 22, k + 24),

in which there are 10 transversal maxima, which is just the number of hori-zontal and vertical series. The symbolic representation of square H10 showsthat

11, 21, 33, 0, 31, 24, 0, 11, 22, 24

are the minimal numbers to be added to the series of the proposed square A,such that a new square is obtained in which maximal terms of the different

10[the explanation of the signs S1, S2 etc. used in the table of square H will the given inthe next paragraph.]


vertical series all belong to different horizontal series and that no other suchsquare may be deduced from A by adding to one of the horizontal series asmaller number that the assigned one.

If the number of quantities of which the squares are made is very great, itwill not be difficult to invent devices by which the trouble of writing numbersbe avoided, for among their great mass, only a few are necessary to form anew square.

[§. 2.We expose the rule for finding minimal numbers `1,`2, . . . , `m, being given an arbitrary system of suchnumbers or being only given the terms of the squaretable that provide the transversal maxima after theaddition of `1, `2, . . . , `m.

An example of the rule is added.]

Again, let ì be positive or null quantities and, having setai,κ + ì = pi,κ,

let the squarep1,1 . . . p1,mp2,1 . . . p2,m

. . . . . . . . . . . .pm,1 . . . pm,m

be such that the maximal terms of the different vertical series also belongto different horizontal series, so that one may find one or more [complete]11

systems of transversal maxima. Distinguishing one of them by asteriks andunderlining the remaining maxima, that are equal to them in each vertical,we shall have this sure criterion by which one may know if such a square isderived from A, which is formed of quantities ai,κ, by adding minimal positiveor null quantities ì to the horizontal series. One takes indeed the horinzontalseries for which ì = 0, or equivalently that are the same than in the proposedsquare A. I shall denote by S1 such series, of which at least one must exist.One takes the underlined terms in the series of S1

12 and the starred terms

11[That is, composed of m terms.]12I call vertical or horizontal series of a term, the horizontal or vertical series in which

it lies. I call transversal terms, terms all placed in different horizontal and vertical series;I simply call maxima of different verticals all placed in different horizontals transversalmaxima. J.

22 C.G.J. Jacobi

in the vertical of these ones; I denote by S2 the horizontal series of thesestarred terms that do not already belong to S1 itself. We take again thestarred terms in the vertical series to which belong the underlined terms ofS2 and I denote by S3 their horizontal series that are different from S1 and S2.If, in this way, we exhaust all the horizontal series, the square formed by thequantities pi,κ is deduced from the proposed one, formed by the quantities ai,κ,by adding minimal positive or null quantities to its horizontal series. So, inour example, all the horizontal series are referred to the systems S1, S2 etc.successively found in the following way:

S1 S2 S3 S4 S5 S6d b a f c eg h k

i

Hence, one may certainly conclude that, in our example, one uses minimalpositive quantities to be added to the horizontal series, in order to build thissolution of the proposed problem.

The method by which the simplest solution may be deduced from an arbitraryone relies on these same principles, by which we obtained a criterion forwhever the problem has been solved in the simplest way, that is by minimalpositive quantities `i. Setting

ai,κ + hi = qi,κ,

where the quantities hi are positive or null, and forming a square of thequantities qi,κ in the same way as the square A is formed of quantities [ai,κ],we assume that one may take maxima in its different series that are also allplaced in different horizontal ones. I denote with asterisks an arbitrary [com-plete] system of such transversal maxima. By subtracting from all the qi,κthe smallest of the quantities hi, which I call h, one produces a square ofwhich one or more horizontal series is unchanged, [i.e.] are the same asin the square A; I denote again these series by S1. Underlining then themaximal terms in their vertical different from the starred ones, one deducessuccessively from the series S1, according to the rule given above, the sys-tems of horizontal series S1, S2, . . . , Sα. If by them we get all the horizontalseries, the simplest solution is found, but if there remains a horizontal se-ries in which there is no starred term being placed in the same vertical asan underlined term of the series S1, S2, . . . , Sα, I subtract from all these se-ries the same quantity h′, the smalest such that one of their starred termsbecome equal to one of the terms of the same vertical belonging to one ofthe series S1, S2, . . . , Sα or such that one of them becomes equal to the corre-sponding series of the square A. So, the number of horizontal series belonging


to the sets S1, S2, . . . , Sα will be made greater than in the square formed ofthe quantities qi,κ − h. Continuing, if needed, this process, the horizontalseries excluded from the sets S1, S2, . . . , Sα will remain fewer and fewer andone comes soon to a square in which the series S1, S2, . . . , Sα contain allhorizontal series.

If, adding some quantities h1, h2, . . . , hm to the horizontal series of the squareA,one gets a square possessing m transversal maxima, the sum of the terms thatoccupy in the square A the same place as these transversal maxima in the de-rived square, will have a maximal value among all the sums of m transversalterms of the square A. Hence the inequality problem,

a square A formed of m2 terms being given, to find m transversal termsof A possessing a maximal sum,

will have as many solutions as one may find systems of transversal maximain the derived square. One finds all these systems if we keep in the derivedsquare only the terms being maximal in their vertical, give all the others anull value and form the determinant of all these terms. Indeed, the differentterms of this determinant give the different solutions of this problem. Onemay reciprocally show that any solution of the foregoing inequality problemgives a system of transversal maxima of the derived square. In our example,one needs to form the determinant of the underlined terms of the square H,the remaining terms of this square being given the value zero. This determi-nant may be successively reduced to the simpler determinants formed of thesquares

I. II. III.

α β δ ζ ι κ

a 102 110b 53 10c 47 49 110e 49 43i 47 49 110k 49 43

,

α δ ζ ι κ

a 102 110c 47 49 110e 49 43i 47 49 110k 49 43

,

α δ ι κ

c 47 49 110e 49 43i 47 49 110k 49 43

.

We indicate here the terms of the square by the indication of the verticaland horizontal series to which they belong, the former denoted by the let-ters a, b, c, etc. and the latter by the letters α, β, γ, etc. In the square H,the terms (d, γ), (g, η) are the only underlined ones in their verticals, theterms (f, ϑ), (g, η), (h, ε), the only underlined ones in their horizontal series.So, all the terms forming the determinant must have the common factor

(d, γ)(h, ε)(g, η)(f, ϑ).

24 C.G.J. Jacobi

This factor eliminated, the determinant formed of the quantities of square Iremains, that arises from the elimination of the horizontal series d, f, g, h andof the verticals γ, ε, η, ϑ. In this square, the term (b, β) is the only non zeroterm in its vertical, so that, removing this common factor, there remains tosearch the determinant of the quantities of square II. In this square again,the term (a, ζ), alone in its vertical, is removed; there remains to form thedeterminant of quantities III

−(c, α)(e, δ)(k, ι)(i, κ)− (i, α)(k, δ)(e, ι)(c, κ)+(c, α)(k, δ)(e, ι)(i, κ) + (i, α)(e, δ)(k, ι)(c, κ)

= −{(c, α)(i, κ)− (i, α)(c, κ)} {(e, δ)(k, ι)− (k, δ)(e, ι)} 13.

As it contains four terms, there will be in the square A four systems oftransversal maxima possessing a maximal sum, viz.

(b, β) + (d, γ) + (h, ε) + (a, ζ) + (g, η) + (f, θ)+ 1) (c, α) + (e, δ) + (k, ι) + (i, κ)or 2) (c, α) + (k, δ) + (e, ι) + (i, κ)or 3) (i, α) + (k, δ) + (e, ι) + (c, κ)or 4) (i, α) + (e, δ) + (k, ι) + (c, κ)

that are numericaly expressed in our example by

32 + 61 + 73 + 91 + 91 + 21 = 369+ 1) 14 + 18 + 19 + 88 = 139or 2) 14 + 25 + 12 + 88 = 139or 3) 25 + 25 + 12 + 77 = 139or 4) 25 + 18 + 19 + 77 = 139,

so that the maximal sum of transversal terms is 508. Reciprocally, if weknow in any way some transversal terms of the proposed square A possessinga maximal sum, one obtains by the addition of minimal quantities `i to thehorizontal series of the proposed square A a square in which all the maximaof the different vertical series are also placed in different horizontal series.

I denote of course with asterisks these given transversal terms possessinga maximal sum and I add to the horizontal series quantities such that theirstarred terms become equal to the maxima of their respective vertical series.I write each augmented series under the remaining ones and compare it tothe remaining ones, the preceding ones and the following ones. For this, Iindicate the horizontal series denoted by the letters a, b, etc. by the sameletters, once the augmentation is made, and I also keep the asterisks of the

13The manuscript gives the result up to the sign.


starred terms. The following table will illustrate this way of proceeding in ourexample. We assume to be given the transversal terms possessing a maximalsum

(a, ζ), (b, β), (c, α), (d, γ), (e, δ), (f, θ), (g, η), (h, ε), (i, κ), (k, ι),91 32 14 61 18 21 91 73 88 19.


(1) a 14 23 1 5 73 91∗ 10 34 5 99(2) b 25 32∗ 2 4 62 81 9 23 4 88(3) c 14∗ 1 7 16 21 7 13 12 3 77(4) d 11 53 61∗ 4 3 1 12 1 4 91(5) e 9 21 23 18 27 3 6 9 12 15(6) f 4 16 18 13 5 12 23 21∗ 14 81(7) g 25 43 13 16 83 10 91∗ 3 7 13(8) h 27 7 17 37 73∗ 8 11 24 23 22(9) i 25 12 18 27 32 18 24 23 14 88∗

(10) k 16 28 30 25 34 10 13 16 19∗ 42

(11) b 46 53∗ 23 25 83 102 30 44 25 109(12) a 25 34 12 16 84 102∗ 21 45 16 110(13) c 46∗ 33 39 48 53 39 45 44 35 109(14) e 39 51 53 48∗ 57 33 36 39 42 45(15) f 28 40 42 37 29 36 47 45∗ 38 105(16) h 38 18 28 48 84∗ 19 22 35 34 33(17) i 47∗ 34 40 49 54 40 46 45 36 110∗

(18) c 47 34 40 49∗ 54 40 46 45 36 110(19) e 40 52 54 49 58 34 37 40 43 46(20) k 40 52 54 49 58 34 37 40 43∗ 66

In the vertical ζ, the starred term is itself maximal, so at first the [hori-zontal series a]14 does not change; in the vertical β, the maximum is 53, sothe horizontal b must be written below, augmented by the number 21, whichforms line (11). Going back to the first term, we find in the series ζ the maxi-mum 102, so a must be increased by 11, which provides line (12). Progressingup to term (c, α), we find in α the maximum 46 placed on line (11), so c mustbe augmented by [32]15, which provides line (13). In the same way, the se-ries d and g remain unchanged, I respectively increase the series e, f, h, i bynumbers 30, 24, 11, 22, which provides lines (14), (15), (16), (17). Then, onefinds in line (17) the term 47 in the vertical α, greater than the starred term

14Jacobi wrote: series horizontalibus non mutatur; I follow Borchardt’s correction: se-ries horizontalis a non mutatur. T.N.

15The manuscript has 13. T.N.

26 C.G.J. Jacobi

of the same vertical 46 placed in line (13), I add 1 to line (13), which providesline (18). In (17) and (18), the term 49 of the series δ is greater than thestarred term of this same vertical, placed in (14), so I increase line (14) itselfby one, which produces line (19). At last, I proceed to the term (k, ι) = 19;and, as the maximum of the vertical ι is 43, placed in (19), I form line (20)by adding 24 to the series k. By this, the work will be achived. We haveindeed found series

a, b, c, d, e, f, g, h, i, k,forming lines (12), (11), (18), (4), (19), (15), (7), (16), (17), (20),

the starred terms of which are maximal in their verticals, as was requested.We see that these series constitute the square H found above by anothermethod.

By using what precedes, one gets a new solution of the problem proposedabove: if the question is, some quantities being added to the horizontalseries of the square A, to get a square, of which the maximal terms in theirverticals all belong to different horizontal series and some such quantitiesare known, how to find minimal ones. For, as according to the assumptionmade, one knows a square derived of A possessing m transversal maxima,one also knows in A m transversal terms possessing a maximal sum. Thesebeing known, following the rule given above, one easily derives from A, bythe addition of minimal positive quantities, a square possessing m transversalmaxima. At the same time, we see how, a system of transversal terms of Apossessing a maximal sum being known, we easily find all the remainingones. For, knowing such a system, we see that one deduces from A a squarepossessing m transversal maxima; in which, if we only keep the maximalterms in each vertical, the remaining being made equal to zero, every non zeroterm of the determinant of the square formed by these quantities providesevery system of transversal maxima, and so every system of transversal termsof the square A possessing a maximal sum; the terms of both systems occupyindeed the same places in the two squares.


[§. 3.The solution of the problem related to a square table of m2

quantities is applied to a system of m differential equa-tions. The normal form or forms to which the proposedsystem may be brought back by a shortest reduction.

Other reductions to normal form.]

The proposed differential equationsu1 = 0, u2 = 0, . . . , um = 0,

had to be respectively differentiated `1, `2, . . . , `m times to be brought backto another system in normal form, by a shortest reduction. The numbers`1, `2, . . . , `m are the same as those whose computation I described above.These being fully determined, the system of auxiliary differential equationsrequested for the shortest reduction, formed of these derivatives, will be alsofixed. But, most of the time, there are many different normal forms towhich the proposed differential equations may be reduced with this systemof auxiliary equations. Again let ai,κ be the order of the highest derivativeof the variable xκ that appears in equation ui = 0 and let us again place thequantities ai,κ into a square A, of which the terms ai,1, ai,2, . . . , ai,m constitutethe ith horizontal series and the terms a1,κ, a2,κ, . . . , am,κ the κ

th vertical. Wetake in the square A an arbitrary system of transversal terms possessing amaximal sum

aα1,1, aα2,2, . . . , aαm,m,

the proposed differential equations may be brought back by a shortest reduc-tion to these, in normal form:

x(aα1,1)1 = X1, x

(aα2,2)2 = X2, . . . , x

(aαm,m)m = Xm,

where the derivatives of the different variables appearing on the left are thehighest that appear in the reduced system, of which the functions X1, X2,. . . , Xm placed on the right may be assumed to be absolutely free. And onewill have as many different such systems, to which the proposed differentialequations may be brought back by a shortest reduction, as one has systemsof transversal terms possessing a maximal sum in the square A.16 These arefound as explained above. Having formed the auxiliary equations used fora shortest reduction, we assume that we find the highest derivative of thevariable xκ in the proposed equations ui = 0, ui1 = 0, etc. or in the auxiliary

16This proposition does not stand in all cases, as shown by the example x′′1 +x′′2 +x

′′3 = 0,

x′2 = 0, x2 + x3 = 0. T.N.

28 C.G.J. Jacobi

equations derived from them by iterated differentiations; in the places of thesquare that belong to the κth vertical series and to the ith, ith1 , etc. horizontalseries, I put a unit or any other [non zero] quantity, and I put zero in theother places of the κth vertical. This being done for each of the variables xκ,I form the determinant of the terms of this square. One of its non-zero term,if it is formed of quantities of the first, second, . . . , mth vertical respectivelybelonging to the α1

th, α2th, . . . , αm

th horizontal series, will give a normal formin which the highest derivatives of the variables x1, x2, . . . , xm are respectivelythe same as in the equations

uα1 = 0, uα2 = 0, . . . , uαm = 0.

As for another term of the determinant one has another succession of theindices α1, α2, . . . , αm, each normal form to which the proposed equationsmay be reduced by a shortest reduction, is given for that reason by eachnon-vanishing term of the determinant.

The method by which, adding minimal positive quantities to horizontal series,one deduces a square in which all the maxima of verticals are placed indifferent horizontal series will be made easier if one knows in some way asystem of m transversal terms of the square possessing a maximal sum. Bythis easier method, one may find how many times each proposed equationmust be differentiated in a shortest reduction in order to form the auxiliaryequations, whenever one will have in any way some normal form to whichthe proposed equations are reduced by such a reduction. Such a normalform is known if the proposed differential equations are such that in eachof them a derivative of a different variable reaches the highest order. For,indeed, these derivatives of the different variables, the highest in the differentproposed equations, will also be the highest in a normal form, to which theproposed differential equations can be brought back by a shortest reduction.One easilly sees that the orders of these derivatives constitute in the square Aa system of m transversal terms.

Let us assume, e.g. that 10 differential equations u1 = 0, u2 = 0, . . . ,u10 = 0 between the independent variable t and the 10 dependent vari-ables x1, x2, . . . , x10 are given and that the numbers of the square A in ourexample indicate the highest orders up to which the derivatives of each depen-dent variable in the different equations go, so that e.g. the highest derivativesof variables x1, x2, . . . , x10 that appear in [equation] u1 = 0 are

x(14)1 , x

(23)2 , x

(1)3 , x

(5)4 , x

(73)5 , x

(91)6 , x

(10)7 , x

(34)8 , x

(5)9 , x

(99)10 .

As the last square H is deduced from the proposed one A by adding to thehorizontal series the numbers


11, 21, 33, 0, 31, 24, 0, 11, 22, 24,

a shortest reduction is obtained with the auxiliary equations formed by dif-ferentiating the proposed equations

u1 = 0, u2 = 0, u3 = 0, u5 = 0, u6 = 0, u8 = 0, u9 = 0, u10 = 0

respectively 11, 21, 33, 31, 24, 11, 22, 24 times, the two equations u4 = 0 andu7 = 0 not being used to form auxiliary equations. With these auxiliaryequations, the proposed equations may be reduced by simple eliminationsto four different normal forms. In all these ones, among the highest deriva-tives that are to be expressed by lower order derivatives and the variablesthemselves, one finds, as explained above

x(32)2 , x

(61)3 , x

(73)5 , x

(91)6 , x

(91)7 , x

(21)8 ;

then in the first normal form: x(14)1 , x

(18)4 , x

(19)9 , x

(88)10 ;

second x(14)1 , x

(25)4 , x

(12)9 , x

(88)10 ;

third x(25)1 , x

(25)4 , x

(12)9 , x

(77)10 ;

fourth x(25)1 , x

(18)4 , x

(19)9 , x

(77)10 .

So, the complete integration of the 10 proposed differential equations re-quires 508 arbitrary constants, this number being the sum of the orders ofthe highest derivatives of the different variables in the normal forms. Allthe highest derivatives appear in the proposed differential equations, butexcept x

(61)3 , x

(91)6 , x

917 , they are not the highest.

We consider an arbitrary reduction and, among all the [auxiliary and proposed]differential equations, we choose the m highest derivatives of the proposedequations; some will be among the proposed equations, if they are not used toform auxiliary equations by differentiation. In each of these m equations, wegather the orders of the highest derivatives of each variable and we disposethem in square in the usual way: it may be proved that in such a square themaxima of the different vertical series are necessarily also placed in differenthorizontal series. And by the rules given above, we can go back from such asquare to some other, deduced from A, by using minimal `i numbers. Hence,from an arbitrary normal form reduction of the proposed differential equation,one may deduce a shortest one.

30 C.G.J. Jacobi

[§. 4.Reduction of the proposed system to a single differentialequation. A rule for finding the reduction is given andillustrated with an example. An elegant form under which

the rule may be expressed.]

A system of differential equations may in general be reducedto a single differential equation in two variables. Let these two vari-ables be the independent variable t and the dependent one x1; thisunique differential equation must be completed by other equations, by whichthe remaining variables are expressed as functions of t, x1 and derivativesof x1, not reaching the order of the differential equation between t and x1.As it is usual that this type of normal form be considered before others bymathematicians, I will indicate how many times each proposed differentialequations u1 = 0, u2 = 0, . . . , um = 0 must be differentiated to produce thedifferential equations necessary for this reduction.

We assume that the proposed differential equations u1 = 0, u2 = 0, . . . ,um = 0 must be respectively differentiated `1, `2, . . . , `m times in orderto obtain the auxiliary equations required for a shortest reduction. I havetaught above how these numbers `1, `2, . . . , `m are found. Adding `1, `2, . . . ,`m to the horizontal series of the square A, I form another square A

′, in whichI distinguish with an asterisk some [complete] system of transversal maximaand I underline the remaining maxima of the various verticals. If all variablesare to be eliminated, except the independent variable t and the dependentvariable xκ, I look for the starred term of the κ

th vertical, which is in the ith

horizontal series; in the ith horizontal series, I look for the underlined terms,in the vertical of each of them, for the starred terms, in the horizontal seriesof which again for the underlined terms, and so on. In this framework, it isuseless to go back again to the starred terms already considered.

Continuing this operation, as far as possible, I will say that all the hori-zontal series to which one comes by this process are attached to the ith fromwhich we have started. I increase these series as well as the ith of a samequantity, the smallest such that one of their terms being neither starred norunderlined becomes equal to a starred term in its vertical. The horizontalseries of this term being added to the series attached to the ith series, I in-crease again the ith series and all these being attached to it, the number ofwhich have just been increased, by the smallest quantity such that one oftheir terms being neither starred nor underlined becomes equal to a starredterm of its vertical; this being done, the number of series attached to the ith


increases again; and so I increase again and again the number of these series,until one comes to a square A′′ in which all horizontal series are attached tothe ith.

I deduce then from A′′ a square A′′′ by increasing [all] the horizontal seriesby a same quantity, such that the term of the ith horizontal series, belongingto the κth vertical be made equal to the greatest sum that a system of mtransversal terms of the square A may have. The numbers by which thehorizontal series of the square A must be increased so that the square A′′′

appears indicate how many times each of the proposed differential equationsmust be differentiated to produce the auxiliary equations needed to get, byeliminations alone, a differential equation between the variables t and xκalone and the other equations by which the remaining variables are expressedas functions of t, xκ and derivatives of xκ.

In our example, A′ is the square denoted by H. We assume that the κth

vertical is the series ζ, whose starred term 102 belongs to the horizontalseries a, in which are placed the underlined terms 84, 45, 110 belongingto the verticals ε, ϑ, κ, whose starred terms belong to the series h, f, i inwhich one has the underlined terms 47 and 49, belonging to verticals α and δ(I do not use 45, because its vertical was already considered); the starredterms of verticals α and δ belong to series c and k, in this last, we have theunderlined term 43, belonging to the vertical ι, whose starred term is placedin e, a series that contains the unique underlined term 49, whose verticalwas already considered. Hence, we find the series attached to a: h, f , i, c,k, e. Increasing all the series a, h, f , i, c, k, e by one , b is added to theseries attached to a, for with this increment, the term 52 of series e or k,belonging to the vertical β becomes 53, which number is equal to the starredterm of the vertical β that belongs to the horizontal b. I increase againthe series a, h, f, i, c, k, e, b by the number 6, this being done, d is added tothe series attached to a; at last, I increase by the number 37 all the seriesexcept g, so that g itself joins the series attached to a. Hence the square A′′

is obtained from the series of A′ or H:a, h, f , i, c, k, e by adding 44from the series b by adding 43from the series d by adding 37,

the series g staying unchanged. As 102 + 44 = 146, 508 − 146 = 362, thehorizontal series of the square A′′ must be increased by the number 362 toobtain A′′′. Denoting, as above, the square A′ by

A′ (a+ 11, b+ 21, c+ 33, d, e+ 31, f + 24, g, h+ 11, i+ 22, k + 24),

we obtain for the squares A′′ and A′′′

A′′ (a+ 55, b+ 64, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68)

32 C.G.J. Jacobi

A′′′ (a+ 417, b+ 426, c+ 439, d+ 399, e+ 437, f + 430, g + 362, h+ 417, i+ 428, k + 430).

So, in our example, to eliminate all variables except t and x6 from the 10proposed differential equations, they must be differentiated respectively 417,426, 439, 399, 437, 430, 362, 417, 428, 430 times to produce the requestedauxiliary differential equations.

By the same method, we obtain the squares A′′, in which all the horizontalseries are respectively attached to each of the series a, b, c, . . . , k, by addingto the series of A′

a, h, f, i, c, k, e, +44; b + 43; d + 37; g 0,b, a, h, f, i, c, k, e, +44; d + 37; g 0,c, k, f, i, e, +44; b, a, h + 43; d + 37; g 0,d, b, a, c, e, f, k, h, i, k, +44; g 0,e, k + 45; b, a, h, f, i, c + 44; d + 38; g 0,f + 44; e, i, c, k + 39; b, a, h, +38; d + 32; g 0g + 9; h + 8; k, e + 7; b, a, f, i, c + 6; d 0,h + 46; k, e + 45; b, a, f, i, c + 44; d + 38; g 0,i, c, k, f, e + 44; b, a, h + 43; d + 37; g 0,k, e + 45; b, a, h, f, i, c + 44; d + 38; g 0.

We see that the third and the ninth, the fifth an the tenth squares are ob-tained from A′ in the same way. We indicate how the squares A′′ are deducedfrom the proposed square A by the following tables:

S. A′′

x6 146 (a+ 55, b+ 64, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68),x2 97 (a+ 55, b+ 65, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68),x1 91 (a+ 54, b+ 64, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68),x3 105 (a+ 55, b+ 65, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68),x9 88 (a+ 55, b+ 64, c+ 77, d+ 38, e+ 76, f + 68, g, h+ 55, i+ 66, k + 68),x8 89 (a+ 49, b+ 59, c+ 72, d+ 32, e+ 70, f + 68, g, h+ 49, i+ 61, k + 63),x7 100 (a+ 17, b+ 27, c+ 39, d, e+ 38, f + 30, g + 9, h+ 19, i+ 28, k + 31),x5 130 (a+ 55, b+ 65, c+ 77, d+ 38, e+ 76, f + 68, g, h+ 57, i+ 66, k + 69),x10 154 (a+ 54, b+ 64, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 54, i+ 66, k + 68),x4 94 (a+ 55, b+ 65, c+ 77, d+ 38, e+ 76, f + 68, g, h+ 55, i+ 66, k + 69).

In the first, second, . . . , tenth horizontal series of the square A′ or H, wehave the starred terms

102, 53, 47, 61, 43, 54, 91, 84, 110, 49,belonging to the

sixth, second, first, third, ninth, eighth, seventh, fifth, tenth, fourthverticals. Adding respectively to these terms

44, 44, 44, 44, 45, 44, 9, 46, 44, 45,the numbers

146, 97, 91, 105, 88, 89, 100, 130, 154, 94,appear, which I have placed, denoted by S, in a marginal column, togetherwith the variables corresponding to the various verticals.


In some square A′′, let S be the starred term of the horizontal series towhich the remaining ones are attached: one will be able to go from S to anyother starred term by a continuous path from a starred term to an underlinedone of the same horizontal and from an underlined term to a starred one ofthe same vertical. We present below, e.g., the first square obtained

A′′ (a+ 55, b+ 64, c+ 77, d+ 37, e+ 75, f + 68, g, h+ 55, i+ 66, k + 68)

or


a 128 146∗ 89 154

b 96∗

c 91∗ 93 89 154

d 98∗

e 96 98 93 87∗

f 91 89∗

g 91∗

h 128∗

i 91 93 89 154∗

k 96 98 93∗ 87

in which I only put the starred terms and the underlined ones, that is thoseequal to the starred terms of the same vertical (omitting to underline them).In this square, all the horizontal series follow from a, the starred term ofwhich is 146. From it, one goes to the other starred terms:

146, 154, 93, 96; 146ζ , 154, 91α; 146, 154, 93, 98; 146, 154, 93, 87;146, 154, 89; 146, 154, 89, 91η; 146, 128; 146, 154; 146, 154, 93.

Two terms T and U placed one after the other are starred terms such thata term from the horizontal series of T , placed in the vertical of U be equalto U , that is underlined, which is the indicated passing rule.

If we suppress from the square A′′ the vertical series of the term S, fromwhich we have started and an arbitrary horizontal, we will easily determinein the remaining square, a system of transversal maxima. We denote by

︷︷TU

34 C.G.J. Jacobi

the term equal to U in the horizontal series of T and placed in the verticalof U and we assume that the starred term of the suppressed horizontal seriesis S(f); so according to the given rule, we go from S to S(f) by the interme-diate starred terms S ′, S ′′, . . . , S(f−1). This being set, the remaining starredterms of the proposed square A′′, will also be the transversal maxima of theremaining square; but instead of S, S ′, . . . , S(i), . . . , one needs to take theterms ︷︷

SS ′,︷︷S ′S ′′,

︷︷S ′′S ′′′, . . . ,

︷︷S(f−1)S(f),

which are equal to S ′, S ′′, . . . , S(f). From this proposition, in the squaresthat remain, having suppressed the vertical series of the term S togetherwith an arbitrary horizontal, the sums of the transversal maxima will be thesame, viz. the sum of the proposed square A′′ decreased by S.

We consider an arbitrary square A′′κ, in which the starred term of the hori-zontal series to which all the remaining ones are attached, belongs to the κth

vertical, which term I denote by Sκ. This square A′′κ is the one that must be

formed when it is proposed to eliminate all variables except t and xκ. Weassume then that the square A′′κ comes from the addition of the quantities

h(κ)1 , h

(κ)2 , . . . , h

(κ)m

to the horizontal series of the square A. We call O the order of the consideredsystem of differential equations, that is the maximal sum of transversal termsin the square A, and let O − Sκ = Pκ; according to the above results onthe formation of auxiliary equations necessary for the proposed elimination,the ith differential equation is to be differentiated Pκ + h

(κ)i times. To which

number Pκ + h(κ)i one may attribute a remarkable meaning. In the square A

′′κ

the sum of the transversal maxima, that is the maximal sum of the transversalterms, is

O + h(κ)1 + h(κ)2 + · · ·+ h(κ)m ,

so, if we suppress the κth vertical series and the ith horizontal, the maximalsum of the transversal terms in the remaining square will be, according tothe proposition found,

O − Sκ + h(κ)1 + h(κ)2 + · · ·+ h(κ)m = Pκ + h

(κ)1 + h

(κ)2 + · · ·+ h(κ)m

and for this reason, if we suppress from the square A the κth vertical seriesand the ith horizontal, the maximal sum of the transversal terms in theremaining square will be Pκ + h

(κ)i . Hence we have found this solution to the

problem enunciated here:


Problem.

Between the independent variable t and the m dependent variables x1,x2, . . . , xm, let there be the differential equations

u1 = 0, u2 = 0, . . . , um = 0;if it is asked to reduce them to a single differential equation between tand xκ, new auxiliary differential equations are to be formed, by dif-ferentiating the proposed differential equations, with the help of whicha differential equation between t and xκ is obtained by simple elimina-tions, without any further differentiation; we search how many timesthe equation ui = 0 must be differentiated to form this system of aux-iliary equations.

Solution.

A square containing m vertical series and as many horizontal series isformed; in the αth vertical and the ath horizontal, we place the order of thehighest derivative of the variable xα that appears in equation ua = 0. Havingsuppressed from this square the ith horizontal series and the κth verticalseries, we look for the maximal sum σi,κ that m− 1 of its terms placed indifferent horizontal series and different vertical series may reach: in order toform the system of auxiliary equations with the use of which the differentialequation between t and xκ will be obtained, the equation ui = 0 must bedifferentiated σi,κ times. The sought number σi,κ will also be equal to theorder of the differential equations that appear if we withdraw ui = 0 fromthe proposed equations and replace xκ by a constant.

The numbers σi,κ = Pκ+h(κ)i = O−Sκ+h

(κ)i are provided by the square A

′′κ,

the construction of which from A′ I explained above. Above I gave the valuesof the numbers Sκ and h

(κ)i corresponding to the proposed example; with

these numbers, one hundred inequality problems are solved, viz. suppressingat the same time from the proposed table a vertical and a horizontal series,to find in the remaining hundred squares the maximal sum of transversalterms. Transversal terms possessing the maximal sum are easily found ineach of these squares, if one goes back to what I explained above about theway of going from a term S of the square A′′ to another arbitrary starredterm S(f) by intermediate starred terms.

36 C.G.J. Jacobi

[§. 5.One determines the condition that lowers the order of the

proposed system of differential equations.]

I t may happen, in some special cases, that the order of the systemsof differential equations do not reach the value of the maximal sumof the transversal terms of the square A. This particular property isindicated by a precise mathematical condition. Again let x

(ai,κ)κ be the highest

derivative of variable xκ that one finds in the equation ui = 0; I form thedeterminant of partial derivatives

∂u1

∂x(a1,1)1

,∂u1

∂x(a1,2)2

, . . . ,∂u1

∂x(a1,m)m

,

∂u2

∂x(a2,1)1

,∂u2

∂x(a2,2)2

, . . . ,∂u2

∂x(a2,m)m

,

. . . . . .∂um

∂x(am,1)1

,∂um

∂x(am,2)2

, . . . ,∂um

∂x(am,m)m

and I only keep its terms

± ∂u1∂x

(a1,i′ )

i′

∂u2

∂x(a2,i′′ )

i′′

· · · ∂u1∂x

(am,i(m)

)

i(m)

in which the sum of orders

a1,i′ + a2,i′′ + · · ·+ am,i(m)

reaches the maximal value O; I suppress all the other terms from the deter-minant. I denote by ∇ the sum of the remaining terms that is in some waya truncated determinant;

∇ = 0

will be the condition for the proposed system of differential equations to havea special structure lowering its order. If ∇ does not vanish, the order ofthe system always reaches the value O assigned by the general theory that Ihave exposed. I call the quantity ∇ the determinant of the proposed systemof differential equations. In our example,

∇ = ∂u1∂x

(91)6

· ∂u2∂x

(32)2

· ∂u4∂x

(61)3

· ∂u6∂x

(21)8

· ∂u7∂x

(91)7

· ∂u8∂x

(73)5

×{

∂u3∂x

(14)1

· ∂u9∂x

(88)10

− ∂u9∂x

(25)1

· ∂u3∂x

(77)10

}{∂u5∂x

(18)4

· ∂u10∂x

(19)9

− ∂u10∂x

(25)4

· ∂u5∂x

(12)9

}.


The four terms of this expression, that appear once the braces expanded,correspond to the four systems of transversal terms of the square A possessinga maximal sum that I have investigated above. So, every time that in ourexample none of the equalities

∂u3

∂x(14)1

· ∂u9∂x

(88)10

− ∂u9∂x

(25)1

· ∂u3∂x

(77)10

= 0,

∂u5

∂x(18)4

· ∂u10∂x

(19)9

− ∂u10∂x

(25)4

· ∂u5∂x

(12)9

= 0,

is satisfied, the system of equations is of order 508, or also their completeintergration requires 508 arbitrary constants. But if one of the two preced-ing equations is satisfied, the order of the system is always lower than thevalue 508. In this case, the proposed differential equations require a prepa-ration that must be made before manipulating them. The non-vanishing ofthe determinant of the proposed differential equations is a condition withoutwhich one cannot deduce the order of the system. Every time the problem ofdetermining transversal terms of the square A possessing a maximal sum hasa unique solution, the order of the proposed system of differential equationsis equal to this maximal sum and it cannot happen that it become smaller.Indeed, the determinant contains a single term and cannot vanish.

the reduction to normal form of a non-normal system of di ...ollivier/production_scient/...reduction...

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